Time Evolution of Schrodinger's Equation

[doublemint]

Hello!

Here is my question:
Consider a particle of mass m, whose initial state has wavefunction $$\psi$$(x), in an infinite potential box of width a. Show that the evolution under the Schrödinger equation will restore the initial state (possibly with a phase factor) after time T=$$\frac{4ma^{2}}{\pi\hbar}$$.

I am not quite sure what to do.
So far i wrote down this:
$$\Psi(x,t) = \psi(x,t) e^{\frac{-iEt}{\hbar}}$$ .. (1)
$$\frac{d\Psi}{dt} = \frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\Psi(x,t)$$ .. (2)
Now subbing (1) into (2),
$$\frac{-iE}{\hbar}\psi(x,t)e^{\frac{-iEt}{\hbar}}=-E\Psi(x,t)$$ using Schrödinger's Time-independent Equation. Where V(x)=0 for a infinite potential well.
Finally: $$\Psi(x,t)=\frac{i}{\hbar}\psi(x,t)e^{\frac{-4iEma^{2}}{\pi\hbar}}$$

I do not think I did it correctly. However its something..
Any help would be appreciated!
Thanks
Shaun

 

[ideasrule]

Write out the stationary states explicitly, in terms of a and m. Do they come back to the initial state after $$\frac{4ma^{2}}{\pi\hbar}$$? They're sine functions, so it should be easy to tell.

 

[doublemint]

I got it! Your ideas rule!